A new approach to the Monge-Amp\`ere eigenvalue problem

TL;DR

This paper introduces a novel method for solving the complex Monge-Ampère eigenvalue problem in hyperconvex domains, establishing uniqueness, providing a formula for eigenvalues, and extending to real Monge-Ampère analogues.

Contribution

The paper presents a new approach based on plurisubharmonic envelopes that simplifies existing methods and extends to complex Hessian and real Monge-Ampère operators.

Findings

Proved uniqueness of eigenfunctions up to positive constants.

Derived a Rayleigh quotient formula for eigenvalues.

Developed an iterative procedure to compute eigenvalues and eigenfunctions.

Abstract

We study the eigenvalue problem for the complex Monge-Amp\`ere operator in bounded hyperconvex domains in $\C^n$, where the right-hand side is a non-pluripolar positive Borel measure. We establish the uniqueness of eigenfunctions in the finite energy class introduced by Cegrell, up to positive multiplicative constants, and provide a Rayleigh quotient type formula for computing the eigenvalue. Under a natural continuity assumption on the measure, we further show that both the eigenvalue and eigenfunctions can be obtained via an iterative procedure starting from any negative finite energy function. Our approach relies on the fine properties of plurisubharmonic envelopes, which allow a partial sublinearization of the nonlinear problem. As far as we know, this method is new, even in the linear case, and not only yields new results but also significantly simplifies existing arguments in the literature. Moreover, it extends naturally to the setting of complex Hessian operators. Finally, by translating our results from the complex Monge-Amp\`ere setting via a logarithmic transformation, we also obtain several interesting analogues for the real Monge-Amp\`ere operator.